Number sets
We extend number sets to solve equations that earlier sets cannot. Complex numbers extend the real numbers by introducing \(i\), where \(i^2=-1\), so that \(x^2=-1\) has solutions \(x=\pm i\).
| Number set | Description | Examples |
|---|---|---|
| Natural \(\mathbb{N}\) | Counting numbers (starting at \(1\) here) | \(1,\ 2,\ 3\) |
| Integers \(\mathbb{Z}\) | Positive and negative whole numbers, and zero | \(-3,\ 0,\ 5\) |
| Rational \(\mathbb{Q}\) | \(\dfrac{p}{q}\), where \(p,q\in\mathbb{Z}\) and \(q\ne0\) | \(-2,\ \dfrac{1}{2},\ 0.75\) |
| Irrational \(\mathbb{R}\setminus\mathbb{Q}\) | Real numbers that are not rational | \(\sqrt{2},\ \pi\) |
| Real \(\mathbb{R}\) | All rational and irrational numbers | \(-2,\ \dfrac{1}{2},\ \sqrt{2}\) |
| Complex \(\mathbb{C}\) | \(a+bi\), where \(a,b\in\mathbb{R}\) and \(i^2=-1\) | \(3,\ 2i,\ 1-2i\) |
Every real number is also complex, since \(a=a+0i\). The sets are related as shown below.
\[\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}\]
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